Tuesday, July 5, 2011

Some Interesting Statistics

If you are not into mathematics, don't worry – you can skip that part and still understand the discussion. But here are a few items that I find somewhat intriguing, and I hope you do too.

The Monty Hall Problem: I discussed this subject on 9 March 2010, so you can read the details there if you wish. Briefly, Monty Hall was the emcee of the game show “Let’s Make A Deal.” The contestant had a choice of three doors to open; behind one was a prize, and goats were hidden behind the other two. After the contestant chose one of the doors, Hall, who knew where the prize was, opened one of the doors behind which was a goat. The contestant was then given a choice: he could either open the door which was his original choice, or he could switch and open the other closed door.
Not surprisingly, most people opted to stay with their original choice. It seems logical that since there are now only two doors from which to choose, there is a 50/50 chance the prize is behind either door.
Unfortunately, that’s wrong! The chances of winning are twice as good if one switches! To see why the apparently illogical choice of switching is actually better, one must understand that before the host opened one of the three doors, the contestant did not know the location of the prize; thus when a door was chosen, he or she had a 1 in 3 chance of being right.
That does not change even after the host opened a door. If the probability of the first door the contestant chose remained the same, and there were only two doors left, that meant either of the remaining unopened doors must have had a 2-in-3 chance of being right — that is, it had twice the chance of holding the prize.

OK – So it doesn't have much practical value, but I think it's an interesting illustration of a counter-intuitive real world event.

Even if one is not a sports fan, everyone has heard of the next item.

Joe DiMaggio Had At Least One Hit In 56 Consecutive Baseball Games: I believe this record will never be broken, in fact, I don't believe any record in any sport will surpass this one. First, the mathematics. This has been figured many ways, but I will use the simplest method I can find.

The probability of any particular event's occurrence is represented by a fraction. For example, the probability of getting heads on a single coin flip is one out of two, or 1/2. The probability of getting heads on two flips in succession is 1/2 x 1/2, or 1/4; three heads in succession is 1/2 x 1/2 x 1/2, or 1/8, a probability of .125, and so on. The odds of three heads in succession are (1-.125)/.125, or 7 to 1 against.
To make it easy, let's assume that DiMaggio came to bat five times in every game. His lifetime batting average was 325. Now you can take my word for this: The probability of his getting at least one hit in five tries was approximately .861. In 1,000 games he would average at least one hit in 861 of them. So his probability of getting at least one hit in 56 consecutive games was .861 x .861 x .861...56 times. That works out to .00022918. Converted to odds, they are 4,362 to 1 against.
With all the baseball games being played, that doesn't really sound like impossible odds, and it isn't. In fact, in a 2008 New York Times op-ed column, Cornell’s Samuel Arbesman and Steven Strogatz wrote that they let a computer replay Major League Baseball’s 135 seasons through 2005, 10,000 times over. And of these simulations of baseball history, 42% of the time there was at least one streak at least as long as DiMaggio’s. Fewer than three times out of 1,000 did the record-holder himself reach that mark. So why do I believe his record will never be broken?
The way I calculated the odds above, I assumed the various games were independent, that is, the outcome of one game did not depend upon any previous games. When one flips a coin, it doesn't matter how many consecutive heads came up previously – the outcome of the next flip is still 1/2. But that's not true of a hitting streak.
Suppose a hitter today gets a long streak going: 20 games, 30 games, etc. As the streak goes on, imagine the mounting pressure he faces. He is constantly being besieged by reporters and fans. He will soon feel that he is letting the team down if he doesn't get a hit. Most importantly, opposing pitchers nibble at the corners of the plate in an effort to get strike calls on questionable pitches. Umpires are under pressure to make the right calls. The longer the streak goes on, the more the pressure mounts.

As I said, it's possible for DiMaggio's streak to be broken, but it would surprise the heck out of me.
Perhaps you can make a little money from my last item.

The Birthday Problem: In a room full of people, what is the probability of finding two who celebrate their birthday anniversary on the same date? Most people have heard that the odds are 50/50 when there are 23 people present, but what if you would like better odds? Here are a few figures:
2 to 1: 29. 3 to 1: 32. 4 to 1: 35. 5 to 1: 36. And the real surprise, if you can find a sucker in a group of 42 people, the odds are 10 to 1 in your favor.
If you really want to know the math, here it is, but be warned, you are going to get into some really, really big numbers:
The probability for a group of size k = 1-(365 x 364 x 363 x ...(365-k+1)/(365 x 365 x 365...k times)).

If you make any money on this item, please remember, you heard it here.

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Each of my next seven chapters will be devoted to exploring one of the categories of jokes. In addition, I shall attempt to give an example or two of jokes which I think fit each category. Introduction - There Are Only Seven Jokes

“There Are Only Seven Jokes” and “The Spirit Runs Through It” and are available in paperback, or at the Kindle Store.